Summation Calculator

About Summation Calculator

Summation (Σ) adds up a sequence of values. Common sums: Σi = n(n+1)/2, Σi² = n(n+1)(2n+1)/6.

How to use this calculator

1. Enter the expression for the term, such as \(i^2\) or \(3i+1\).
2. Set the lower limit \(a\), which is where the index starts.
3. Set the upper limit \(b\), which is where the index stops.
4. Click calculate to find the total of all terms from \(i=a\) to \(i=b\).

The formula explained

The formula \(\sum_{i=a}^{b} f(i)\) computes the total of the values of \(f(i)\) for every integer index from \(a\) through \(b\). Each term is evaluated, then all the results are added together.

• i = the index of summation, a counting variable that changes from \(a\) to \(b\)
• a = the starting value of the index
• b = the ending value of the index
• f(i) = the expression or rule for each term in the sum

Step by step method

1. Substitute each integer value of \(i\) from \(a\) to \(b\) into the expression \(f(i)\).
2. Compute every term one by one.
3. Add all the terms to get the final sum.

Worked example

Problem. Find the sum \(\sum_{i=1}^{4} (2i+1)\).

1. Evaluate the terms: when \(i=1\), \(2(1)+1=3\); when \(i=2\), \(2(2)+1=5\); when \(i=3\), \(2(3)+1=7\); when \(i=4\), \(2(4)+1=9\).
2. Add the values: \(3+5+7+9\).
3. The total is \(24\).

Tips and common mistakes

• Make sure the index starts at the lower limit and ends at the upper limit, because skipping a value changes the total.
• Watch parentheses carefully, especially in expressions like \((i+1)^2\) versus \(i+1^2\), since they give different results.

Frequently asked questions